Pole placement problem for singular systems. (Q2889386)

The paper is meant as a continuation of the work [\textit{M. Dodig} and \textit{Stošić}, Linear Algebra Appl. 431, No. 8, 1267--1292 (2009; Zbl 1170.93016)] and its main goal is to solve the problem of pole structure assignment by state feedback in linear systems. Such a problem is one of the key problems of linear control theory to which a great deal of attention has been paid up to now. The paper provides a complete solution to the problem of pole structure assignment, i.e. it extends the results obtained in [\textit{J. J. Loiseau} and \textit{P. Zagalak}, ``On pole structure assignment in linear structure'', preprint] where just necessary conditions (that are also sufficient in some special cases) of solvability are stated. The main vehicle to achieving this result is the result obtained by the author in [\textit{M. Dodig} and \textit{Stošić}, Linear Algebra Appl. 431, No. 8, 1267--1292 (2009; Zbl 1170.93016)], which is in fact more general than that stated in this paper. This is also mentioned by the author in thr introduction. The results stated in [\textit{M. Dodig}, Linear Algebra Appl. 428, No. 1, 259--304 (2008; Zbl 1131.15010)] and [\textit{M. Dodig} and \textit{Stošić}, Linear Algebra Appl. 431, No. 8, 1267--1292 (2009; Zbl 1170.93016)] are indeed very general and it is very difficult to figure up their impact upon the fields of possible applications. From that point of view, it would be nice if the results of the last mentioned paper could be restated in a form that would give a somewhat simpler approach to the underlying problem and provide, at least partially, a bridge between the approaches used in the last mentioned paper, [\textit{J. J. Loiseau} and \textit{P. Zagalak}, ``On pole structure assignment in linear structure'', preprint] and [\textit{M. Dodig}, Linear Algebra Appl. 428, No. 1, 259--304 (2008; Zbl 1131.15010)].NEWLINENEWLINEI expected that the author would use the methods of pole structure assignment for this purpose. Unfortunately, the paper represents just a little progress in that direction. One of the main drawbacks of the paper is a lack of examples that would provide the reader with a deeper insight into the feedback invariants and their role, and give a hint how to proceed when constructing the feedback(s) that assigns the desired pole structure. The literature devoted to pole/pole structure assignment by state feedback is undoubtedly very extensive and this fact could be better reflected by the paper. Despite the above criticism, the results introduced in the paper are interesting and can motivate specialists in the field to promote further research.